Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the quadratic equation by completing the square. Verify your answer graphically. $$x^{2}+8 x+14=0$$

Step-by-Step Solution

Answer

The solution to the quadratic equation \(x^2 + 8x + 14 = 0\) by completing the square are x = -4 + √2 and x = -4 - √2.

Step 1: Rewrite in Completing the Square Form

The equation \(x^2 + 8x + 14 = 0\) can be rearranged as \(x^2 + 8x = -14\)

Step 2: Complete the Square

To complete the square, take half the coefficient of x, square it and add it to both sides of the equation. The coefficient of x (from \(8x)\) is 8, half of it is 4, and square of 4 is 16. So, \(x^2 + 8x + 16 = -14 + 16 \Rightarrow (x+4)^2 = 2.

Step 3: Solve for x

Take the square root of both sides, remembering that there could be a positive or a negative solution: \(x+4 = \sqrt{2}\) and \(x+4 = -\sqrt{2}\). Solve these to find x: \(x = -4 + \sqrt{2}, -4 - \sqrt{2}\).

Step 4: Verify the Solution Graphically

Graph the function \(y = x^2 + 8x + 14\) and check if it intersects the x-axis at \(x = -4 + \sqrt{2}\) and \(x = -4 - \sqrt{2}\). If it does, the solutions are verified.

Key Concepts

Completing the Square

When solving quadratic equations, completing the square is a method that can help transform a quadratic equation into a form that is easier to solve. Let's explore this method step by step. Here, our original equation is \(x^2 + 8x + 14 = 0\). Our goal is to rewrite it as a perfect square trinomial, which generally takes the form \((x + p)^2 = q\).

• First, move the constant term to the right: \(x^2 + 8x = -14\).
• Next, identify the coefficient of \(x\) (which is 8), take half of it (4), and square it (16).
• Add this square both to the left side and the right side, transforming the equation: \(x^2 + 8x + 16 = -14 + 16\).
• Now, the left side is a perfect square: \((x+4)^2 = 2\).

This format allows us to easily find the solutions for \(x\) by taking the square root of both sides.

Graphical Verification

Graphical verification is a fascinating way to confirm solutions to an equation using a graph. Here, once we have found the solutions by completing the square, we can visually check them on a graph of the quadratic function related to the equation. In this case, that function is \(y = x^2 + 8x + 14\).
By graphing this function, we look for the points where it intersects the x-axis. These intersection points, known as roots or zeros, should match our solutions. For this problem, we already found the solutions \(x = -4 + \sqrt{2}\) and \(x = -4 - \sqrt{2}\).

• Plot the graph of \(y = x^2 + 8x + 14\).
• Examine where the curve crosses the x-axis.
• The x-coordinates of these intersection points should match our calculated solutions.

Using this visual method not only confirms that the solutions are correct but also enhances your understanding of how algebraic solutions relate to graphical representations.

Solving Quadratic Equations Step by Step

Solving quadratic equations can initially seem daunting, but breaking it down step by step can make it manageable and even intuitive. Let’s go through the journey of solving \(x^2 + 8x + 14 = 0\) using the method of completing the square from the beginning.

• Step 1: Rearrange the equation to isolate the quadratic terms. For our equation, it's rearranged to \(x^2 + 8x = -14\).
• Step 2: Complete the square by adding the square of half of \(x\)'s coefficient to both sides. Add 16: \(x^2 + 8x + 16 = 2\).
• Step 3: Express the left side as a square trinomial: \((x+4)^2 = 2\).
• Step 4: Solve for \(x\) by taking the square root of both sides and solving the resulting linear equations: \(x+4 = \pm \sqrt{2}\).
• Step 5: Simplify to get the solutions: \(x = -4 + \sqrt{2}\) and \(x = -4 - \sqrt{2}\).

By following these steps, solving any quadratic equation using the completing the square method becomes straightforward. Understanding each step ensures you can confidently apply this technique to any similar problem.